If the sum of two positive numbers p and q is 4, find the minimum possible value of the expression (p+1/p)2+ (q+1/q)2

Sure, let's solve this step by step:

Step 1: We know that p + q = 4. Since p and q are positive numbers, we can say that p, q > 0.

Step 2: We need to find the minimum value of the expression (p+1/p)² + (q+1/q)².

Step 3: Let's simplify the expression. We can rewrite it as (p²+2+1/p²) + (q²+2+1/q²).

Step 4: Now, we can apply the AM-GM inequality (Arithmetic Mean-Geometric Mean Inequality) which states that the arithmetic mean of non-negative real numbers is greater than or equal to the geometric mean.

Step 5: Applying AM-GM to p² and 1/p², we get (p²+1/p²)/2 ≥ √(p²*1/p²) = 1. So, p²+1/p² ≥ 2.

Step 6: Similarly, applying AM-GM to q² and 1/q², we get (q²+1/q²)/2 ≥ √(q²*1/q²) = 1. So, q²+1/q² ≥ 2.

Step 7: Therefore, the minimum value of the expression (p+1/p)² + (q+1/q)² = (p²+2+1/p²) + (q²+2+1/q²) is 2+2+2+2 = 8 when p = q = 2.